Divisorial domains

Bazzoni, Silvana

doi:10.1515/form.2000.011

Cited by 26 publications

(21 citation statements)

References 22 publications

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“…The class of domains in which each nonzero ideal is divisorial was studied, independently and with different methods, by Bass [10], Matlis [27], and Heinzer [19] in the 1960s. Following Bazzoni and Salce [12,11], these domains are now called divisorial domains. Among other results, Heinzer proved that an integrally closed domain is divisorial if and only if it is a Prüfer domain with certain finiteness properties [19,Theorem 5.1].…”

Section: Introductionmentioning

confidence: 99%

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Mimouni¹

2016

Turk J Math

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Section: Introductionmentioning

confidence: 99%

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Mimouni¹

2016

Turk J Math

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“…(1) It is well known (see [15] or [11]) that a divisorial domain is h-local and locally divisorial, thus by Theorem 1.4(2) we may assume that R is local. Since R is divisorial, R/I is a module satisfying the AB-5 * condition for every non-zero ideal I of R (see [5,Sec. 2]).…”

Section: S Bazzoni and L Salcementioning

confidence: 99%

Almost perfect domains

Bazzoni

Salce

2003

Colloq. Math.

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Abstract. Commutative rings all of whose quotients over non-zero ideals are perfect rings are called almost perfect. Revisiting a paper by J. R. Smith on local domains with TTN, some basic results on these domains and their modules are obtained. Various examples of local almost perfect domains with different features are exhibited. Introduction.A commutative ring R with 1 is called almost perfect if every quotient of R over a non-zero ideal is a perfect ring. In a recent paper [6] we characterized the commutative integral domains R which are almost perfect as those domains such that all R-modules have a strongly flat cover, or equivalently, such that all flat R-modules are strongly flat. The last property amounts to saying that weakly cotorsion modules (that is, cotorsion in Matlis' sense) are cotorsion (in Enochs' sense).The goal of this paper is to investigate more deeply almost perfect commutative rings. In the local case, almost perfect domains have already been introduced by J. R. Smith in [17] under the name of domains with TTN. We will reconsider here some of his results. We concentrate on almost perfect domains, because we show in Section 1 that an almost perfect ring that is not a domain is perfect.J. R. Smith [17] proved that if every torsion module over a domain R is semi-artininan, then R is locally almost perfect. An important property is missing in order to prove the converse, namely, h-locality (see [6]). We show that a classical example of almost Dedekind domain constructed by , which fails to be h-local, has all its torsion modules semi-artinian. If R is a local almost perfect domain and Q denotes its field of quotients, we prove that the Loewy length of Q/R equals ω if and only if the maximal ideal of R is almost nilpotent.In Section 3 we exhibit three different types of construction of almost perfect local domains. The first construction enables us to provide an example of an integrally closed local almost perfect domain which is not a

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“…A complete characterization of divisorial domains was obtained only very recently by Bazzoni [3]. But a deep knowledge of them, at least in some important cases, was available already in the '60's.…”

Section: Torsionless and Divisorial Domainsmentioning

confidence: 99%

“…Recently the characterization of arbitrary divisorial domains have been obtained by Bazzoni [3], who connected this property with the AB-5 * condition, dual of the Grothendieck condition AB-5 for abelian categories. We omit the technical definition of this property, but we recall that the socle S of a module M is the sum of all simple submodules of M , and that S is essential in M if all non-zero submodules of M intersect S non-trivially.…”

Section: An Integrally Closed Domain Is Divisorial If and Only If Imentioning

confidence: 99%

Warfield Domains: Module Theory from Linear Algebra to Commutative Algebra Through Abelian Groups

Salce

2002

Milan Journal of Mathematics

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In this survey paper we illustrate the generalizations to modules over integral domains of the basic result of linear algebra stating that every finite dimensional vector space is canonically isomorphic to its bidual. These generalizations culminate in the discovery of Warfield domains. We present the characterization of Warfield domains in terms of properties of their overrings, as well as their intrinsic characterizations in the classical cases of Noetherian and integrally closed domains, and in the general case. Finally, the development of the results on direct sum decompositions of torsionless modules from the classical case of Dedekind domains up to Warfield domains is sketched; this subject was originated by the 1912 result of Steinitz on finite direct sums of ideals in algebraic number rings.

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Divisorial domains

Cited by 26 publications

References 22 publications

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Almost perfect domains

Warfield Domains: Module Theory from Linear Algebra to Commutative Algebra Through Abelian Groups

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